Spectral theory is concerned with properties of linear operators in Banach spaces.

Given a linear operator in a complex Banach space with domain the *spectrum* is the set of all complex such that is **not** an isomorphism (here is equipped with the graph norm). Most prominent and already known from finite dimensions (linear algebra) are *eigenvalues* , but in infinite dimensions new phenomena arise.

The complement of in is called the *resolvent set of *.

Central topics in spectral theory are properties of the spectrum including investigation of eigenvalues and eigenvectors, properties of the *resolvent map* , decomposition of the space in invariant subspaces and existence of functional calculi for .

In the lecture we shall study in particular

- spectrum and resolvent for bounded and unbounded operators,
- spectral properties of compact operators in Banach spaces,
- the spectral theorem for self adjoint operators in Hilbert space,
- applications to differential operators and boundary value problems.

# Summary

Summary of the lectures |

# References

J.B. Conway: A Course in Functional Analysis, Springer.

E.B. Davies: Spectral Theory and Differential Operators, Cambridge University Press.

N. Dunford, J.T. Schwartz: Linear Operators, Part I: General Theory, Wiley.

D.E. Edmunds, W.D. Evans: Spectral Theory and Differential Operators, Oxford University Press.

T. Kato: Perturbation Theory of Linear Operators, Springer.

S. Lang: Real and Functional Analysis, Springer.

R. Meise, D. Vogt: Introduction to Functional Analysis, Oxford, Clarendon Press.

W. Rudin: Functional Analysis, McGraw-Hill.

D. Werner: Funktionalanalysis, Springer.